素数判定 费马测试

;; Two numbers are said to be congruent modulo N if they both

;; have the same remainder when divided by N

;; The Fermat test:

;; Given a number N, pick a random number A < N

;; and compute the remainder of A^N modulo N.

;; If the result is not equal to A, then N is certainly not prime

;; If it is A, then chances are good that N is prime.

;; Now pick anthor random number A and test it with the same method.

;; If it also satisfies the equation,

;; then we can be even more confident that N is prime.

;; By trying more and more values of A,

;; We can increase our confidence in the result.

;; This algorithm is known as the Fermat test.

( define ( bad-exp base exp )

( define ( bad-expmod base exp m )

( define ( expmod base exp m )

;; 第一个过程比第二个过程慢,且超过一定长度的数值就不能计算,

;; 因为第一个是先将阶乘算出来，再取模，而第二个是在阶乘的过程中就取模

;; 26 mod 4 和 ( 2 mod 4 ) * ( 13 mod 4 ) 结果一样

;; ( expmod 32 10911110033 10911110033 )

;; ( my-expmod 32 1091111003 1091111003 )

( define ( fermat-test n )

( define ( fast-prime? n times )

;; The Fermat test differs in character from most familiar algorithms,

;; in which one computes an answer that is guaranteed to be correct.

;; Here, the answer obtained is only probably correct.

;; More precisely, if N ever fails the Fermat test,

;; we can be certain that N is not prime.

;; But the fact that N passes the test,

;; while an extremely strong indication,

;; is still not a guarantee that N is prime.

;; What we would like to say is that for any number N,

;; if we perform the test enough times and find that N always passes the test,

;; then the probability of error in our primality test can be made as small as we like.

;; Unfortunately, this assertion is not quite correct.

;; There do exist numbers that fool the Fermat test:

;; numbers N that are not prime and yet have the property that

;; an is congruent to a modulo n for all integers A < N.

;; Such numbers are extremely rare,

;; so the Fermat test is quite reliable in practice

;; There are variations of the Fermat test that cannot be fooled.

;; In these tests, as with the Fermat method,

;; one tests the primality of an integer N

;; by choosing a random integer A < N and

;; checking some condition that depends upon N and A.

;; On the other hand, in contrast to the Fermat test,

;; one can prove that, for any N,

;; the condition does not hold for most of the integers A < N

;; unless N is prime.

;; Thus, if N passes the test for some random choice of A,

;; the chances are better than even that N is prime.

;; If N passes the test for two random choices of A,

;; the chances are better than 3 out of 4 that N is prime.

;; By running the test with more and more randomly chosen

;; values of A we can make the probability of error as small as we like.

;; The existence of tests for which one can prove that

;; the chance of error becomes arbitrarily small has sparked interest in

;; algorithms of this type,

;; which have come to be known as probabilistic algorithms.

;; There is a great deal of research activity in this area,